
Big Bang Nucleosynthesis includes constraining the number of the excited relativistic degrees of freedom. Traditionally, this procedure is interpreted in terms of a bound on the effective number of neutrino. In a more general way, this bound can be translated into a limit on dark radiation. By dark radiation, it is supposed extra massless or approximately massless degrees of freedom while early Universe expanded[1]}. In the case of the model, new particles have mass in range \(\rm M\sim 10^{-3}-10^{-1}~\rm GeV\)  and so this means the parameters space is not sensitive to Big Bang Nucleosynthesis.

The observation of cosmic charge leptons have reported in wide range of the data (\(10~\rm GeV\) -\(10~\rm TeV\) ); Fermi-LAT has reported a measurement of the CR electron-positron spectrum from \(7~\rm GeV\)  to \(2~\rm TeV\)  [2]}. The PAMELA satellite
experiment[3]} released an abundance of the positron in the CR energy range of \(15-100~\rm GeV\) . Also the results of a CR electron-positron spectrum, between \(10~\rm GeV\)  and \(\rm 3 TeV\) , have been reported based upon observations with CALET[4]}. The DAMPE collaboration [5]} has been presented a measurements of the electron-positron spectrum in the energy range \(25~\rm GeV\)  to \(4.6 ~\rm TeV\) . These observations are both considered as signs of standard astrophysical sources or DM annihilation. In our model, DM has mass smaller than \(10~\rm GeV\)  then the parameter space is not sensitive to these observations.

